From a jaded point of view it can be argued that the so called "Special Relationship" between the USA and UK seems to be either non-existent, or just a sop to UK politicians when acting as Washington's poodle, so what does it take to break this bad relationship?
Bad relationship examples for the UK would include; forcing the sale of British assets during WWII; withholding atomic bomb information post war; encouraging the end of British influence post war; threatening to destabilise the UK economy in reaction to Suez; discouraging the development of a British space programme; dragging the UK into numerous wars, Korea, Iraq, Afghanistan, and trying to with Vietnam.
Your aim is not for the UK to become an SSR light - left wing socialist workers paradise - just a functioning Western democracy which has a 'proper grown up relationship' with the USA - something like Germany, Australia, or even France.
So what does it take for the UK government to finally declare the "Special Relationship" over? When is enough, enough? Bonus points for plausibility.
The easiest is probably an overreaction by Eisenhower:
In response to Suez the US Government sells a large part of its Sterling Bond holdings straight away, with a threat to sell more if Britain doesn't pull out - Eisenhower wants to show the UK he is serious. The sale cause a collapse of the pound and seriously damages the UK economy - as it becomes known that the US has threatened to sell further holdings the effects are magnified. There are rolling power cuts and rationing returns as fuel and food imports are reduced. Eden resigns due to ill health and an emergency election is called.
The conservatives under Macmillan manage to win, after a quick campaign calling for national unity in the face of US bullying. The Special Relationship is declared over and the future is declared to be a strong partnership with Europe. The UK joins the EEC in late 1957, and the economy recovers quickly aided by a rapid growth of exports fuelled by a cheaper pound. Future British foreign policy is now carried out with a European rather than Atlantic frame of reference, relations with the US recover to become cordial at best.